Please help with fmincon function

Hi,

I am trying to estimate parameters of an arch model using MLE.

If my code works fine I should get estimates 0.1 and 0.4 ( which I used to create the data)

However, I am having some issues and the estimates I get are wrong.

Can someone help me with this please. Given below are the codes I used.

% data generation

clc;
clear;

p=1;
T = 3000;

ra = zeros(T+2000,1);
seed=123;
rng(seed);

ra = randn(T+2000,1); 

epsi=zeros(T+2000,1);
simsig=zeros(T+2000,1);

a0=0.1; a1=0.4;
unvar = a0/(1-a1);

for i = 1:T+2000
                 if (i==1) 
                     simsig(i) = a0+a1*((a0)/(1-a1));
                     s=(simsig(i))^0.5;
                     epsi(i) = ra(i) * s;

                   else                     
                       simsig(i) = a0+ a1*(epsi(i-1))^2;
                       s=(simsig(i))^0.5;
                       epsi(i) = ra(i)* s;
                   end
  end

yt = epsi(2001:T+2000);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ESTIMATION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

theta0 = [1;1];
A=[0 1];
b=0.99999;

[theta, opt] = fmincon(@(theta)...
               lach(theta,yt),theta0,A,b,[],[],0,1);

% function lach

function L = lach(theta,y)

w     =  theta(1); 
alpha =  theta(2);

y2   = y.^2;
[T1,K] = size(y2); 
ht    = zeros(T1,1);
ht(1) = sum(y2)/T1;

for i=2:T1
    ht(i)=w + alpha*y2(i-1);
end

sqrtht  = sqrt(ht);
x       = y./sqrtht;

l = -0.5*log(2*pi) - log(sqrtht) - 0.5*(x.^2);
l=-l;
L = sum(l);

Also, if you can kindly explain the difference between the following two ways of calling fmincon, that will be highly appreciated.

[theta, opt] = fmincon(@(theta)...
               lach(theta,yt),theta0,A,b,[],[],0,1);

[theta, opt] = fmincon(@lach,theta0,A,b,[],[],0,1);

thanks

• When writing these codes I got some help from codes I found online.

[dav]

Output of my code:

Local minimum possible. Constraints satisfied.

fmincon stopped because the size of the current search direction is less than
twice the default value of the step size tolerance and constraints are 
satisfied to within the default value of the constraint tolerance.

<stopping criteria details>

No active inequalities.
>> theta

theta =

      0.0002
    -13.4557

[the cyclist]

I suggest you edit your question to give it a more meaningful title that might catch the eye of someone who knows about these functions.

[dav]

thanks

[Matt J]

If my code works fine I should get estimates 0.1 and 0.4 ( which I used to create the data)

What happens when you run fmincon with 0.1 and 0.4 as your initial guess? What is the final value for opt in that case and what was it for

theta =
      0.0002
    -13.4557

Answer 1

[Walter Roberson]

[theta, opt] = fmincon(@lach,theta0,A,b,[],[],0,1);

will not have any way of getting y into the routine lach. What you have now,

[theta, opt] = fmincon(@(theta) lach(theta,yt),theta0,A,b,[],[],0,1);

is fine.

The message "Local minimum possible. Constraints satisfied." is a message you would get if it was successful in finding a minimum. fmincon cannot tell whether the minimum it finds is a local minimum or the global minimum so it is being conservative in warning you that what it has find might be a local minima.

The message "fmincon stopped because the size of the current search direction is less than" and so on, is less common. You are getting it because you are hitting against the minimum for the first pararameter. The algorithm has detected that it cannot take two more steps towards the minimum, using the minimum step size that it has been configured to use. You could pass an options structure to specify a smaller minimum step size, but the effect of that would just be to get you ever closer to the boundary 0.

I suspect that you are running into a local minimum. That happens with fmincon(). You can try experimenting with different theta0 (different starting points.) If you had the global optimization toolbox, you could use a routine that would run fmincon() with a variety of different starting points, automatically, looking for the best result out of the set. The global optimization toolbox also has other minimization routines that can be used.

[dav]

Thank you very much for the detailed description.

when I used some of the theta0 s it takes very long time to run or doesnt end at all.

Also how is it possible to get negative values when I give the lower bound to be 0?

Is it possible to use fminsearch with constraints? because when i used it without constraints most of the times I got better answers; but I cannot use negative or greater than one parameter estimates.

thanks again

[Matt J]

You passed scalars for ub and lb whereas your theta is a 2x1 vector. Therefore, only the first component was bounded.

[dav]

thanks alot!

Can you please tell me if i can use fminsearch with constraints ? i get better answers from that!

[Walter Roberson]

No, fminsearch does not deal with constraints.

Have you tried telling fmincon to use a different algorithm? http://www.mathworks.com/help/optim/ug/choosing-a-solver.html#bsbqd7i

[Matt J]

Can you please tell me if i can use fminsearch with constraints ?

There are ways to extend fminsearch to deal with constraints, e.g.,

http://www.mathworks.com/matlabcentral/fileexchange/8277-fminsearchbnd-fminsearchcon

However, it is not rational that fminsearch does better than fmincon. You might be blaming witches for the crops.

[dav]

Thanks a lot for all your help!